Finite Group Databasebeta

← #17978409

Subgroups — #17978409

Order 4352.

706 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763868C212yesmaximal
2#14545121C212yesmaximal, complemented
3#14545120C214yesmaximal, complemented
4#14545117C212yesmaximal, complemented
5#14545116C211yesmaximal, characteristic, complemented
6#14545114C212yesmaximal, complemented
7C2×C4⋊#63216C211yesmaximal, characteristic, complemented
8C4×#10587593C211yesmaximal, characteristic
9#11763868—28no—
10C2×#10587530C2214yes—
11C2×#10587538C2212yes—
12C2×#10587529C2214yes—
13C22×#63216C2212yes—
14#11763730C2214yes—
15#11763730C2214yes—
16C4⋊#63216C2211yescharacteristic
17C4⋊#63216C2211yescharacteristic
18C4⋊#63216C2212yes—
19C4⋊#63216C2212yes—
20C4⋊#63216C2212yes—
21C2×#10587593C411yescharacteristic
22C2×#10587593C2211yescharacteristic
23C4×#63252C411yescharacteristic
24#11762767C411yescharacteristic
25#11762767C2211yescharacteristic
26#11762766C412yes—
27#11762766C2212yes—
28C4×#63218C412yes—
29C2×C4×#3622C2211yescharacteristic
30C4×C8×D17C2212yes—
31C4×M16×C17C411yescharacteristic
32C2×#63252C2×C411yescharacteristic
33C2×M16×C17C2×C411yescharacteristic
34C2×M16×C17C2×C411yescharacteristic
35C2×C42×C17C2×C411yescharacteristic
36C4×C22⋊D17C812yes—
37C4×C22⋊D17C812yes—
38#10587612C2×C411yescharacteristic
39C22×#3622C2×C411yescharacteristic
40C22×#3622C2311yescharacteristic
41#10587593—24no—
42#10587593D414yes—
43#10587593Q814yes—
44C2×#63218C2×C412yes—
45C2×#63218C2×C412yes—
46C2×C8×D17C2×C412yes—
47C2×C8×D17C2312yes—
48C4×D68C811yescharacteristic
49C4×D68C811yescharacteristic
50#10587558C2×C412yes—
51C42×D17C2×C412yes—
52C4×Dic34C811yescharacteristic
53C4×Dic34C811yescharacteristic
54#10587538—24no—
55C2×#63216—24no—
56C2×#63216C2312yes—
57C2×#63216C2312yes—
58C2×#63216C2314yes—
59#10587530—28no—
60#10587529—28no—
61C17×#97C2×C412yes—
62C4×C8×C17C2×C412yes—
63#10587486C2×C412yes—
64#10587486C2×C412yes—
65C8×Dic17C2×C412yes—
66C8×Dic17C2312yes—
67C4×#3622C2×C411yescharacteristic
68C4×#3622C2×C411yescharacteristic
69C4×#3622C2311yescharacteristic
70C4×#3622C2311yescharacteristic
71C22×C4×C17C2×C811yescharacteristic
72C22×C4×C17C2×C811yescharacteristic
73C22×C4×C17C22×C411yescharacteristic
74C2×C22⋊D17C2×C812yes—
75#63252C4⋊C414yes—
76#63252M1614yes—
77C2×D68C2×C811yescharacteristic
78C2×C4×D17C2×C812yes—
79C2×C4×D17C2×C812yes—
80C2×C4×D17C22×C412yes—
81C2×Dic34C2×C811yescharacteristic
82M16×C17—22no—
83M16×C17C4⋊C414yes—
84C2×C8×C17C22×C412yes—
85C2×C8×C17C22×C412yes—
86C42×C17C2×C811yescharacteristic
87C42×C17C2×C811yescharacteristic
88C42×C17C22×C411yescharacteristic
89C42×C17C22×C411yescharacteristic
90C22⋊Dic17C2×C812yes—
91#63227C2×C812yes—
92#63227C2×C812yes—
93C4⋊Dic17C2×C811yescharacteristic
94C4⋊Dic17C2×C811yescharacteristic
95#63225C2×C812yes—
96#63225C2×C812yes—
97C4×Dic17C2×C812yes—
98C4×Dic17C2×C812yes—
99C4×Dic17C22×C412yes—
100C2×#3622—22no—
101C2×#3622C22×C411yescharacteristic
102C2×#3622C22×C411yescharacteristic
103C2×#3622C22×C411yescharacteristic
104C2×#3622C2×D412yes—
105C2×#3622C2×Q812yes—
106C2×#3622C2411yescharacteristic
107#63218—24no—
108#63218C4⋊C418yes—
109C8×D17—24no—
110C8×D17C2×D414yes—
111C8×D17C2×Q814yes—
112#63216—22no—
113#63216—24no—
114#63216—22no—
115#33398—171nomaximal, Sylow
116C23×C17C22×C811yescharacteristic
117C22×D17C22×C812yes—
118C2×C4×C17—22no—
119C2×C4×C17—41no—
120C2×C4×C17C2×C4⋊C412yes—
121C2×C4×C17C22×C811yescharacteristic
122C2×C4×C17C22×C811yescharacteristic
123C2×C4×C17C22×C811yescharacteristic
124C2×C4×C17C22×C811yescharacteristic
125C2×C4×C17C22×C811yescharacteristic
126C2×C4×C17C2×M1612yes—
127C2×C4×C17C23×C411yescharacteristic
128C22⋊D17C4⋊C818yes—
129C2×Dic17C22×C812yes—
130C2×Dic17C22×C812yes—
131C2×Dic17C22×C812yes—
132D68C4⋊C814yes—
133C4×D17—24no—
134C4×D17—42no—
135C4×D17C2×C4⋊C414yes—
136C4×D17C2×M1614yes—
137Dic34C4⋊C814yes—
138C8×C17—22no—
139C8×C17C2×C4⋊C414yes—
140#3622—21no—
141#3622—21no—
142#3622C2×C4⋊C412yes—
143#3622C22×D411yescharacteristic
144#3622C22×Q811yescharacteristic
145C2×#504—172no—
146C4×#131—171no—
147#2151—174no—
148#2151—172no—
149#2150—171no—
150#2150—172no—
151C2×C4⋊C16—171no—
152C2×C4⋊C16—172no—
153C22×C17—21no—
154C22×C17—21no—
155C22×C17C2×C4⋊C812yes—
156C22×C17C23×C811yescharacteristic
157C2×D17—22no—
158C2×D17—22no—
159C2×D17C2×C4⋊C814yes—
160C4×C17—21no—
161C4×C17—21no—
162C4×C17—21no—
163C4×C17C2×C4⋊C812yes—
164C4×C17#50412yescomplemented
165C4×C17C22×C4⋊C411yescharacteristic
166C4×C17C22×M1611yescharacteristic
167Dic17—22no—
168Dic17—22no—
169Dic17C2×C4⋊C814yes—
170C2×#131—171no—
171C2×#131—171no—
172C4×#41—171no—
173#504—348no—
174C2×M32—172no—
175C2×M32—174no—
176C22×C16—172no—
177C22×C16—174no—
178#405—171no—
179#405—171no—
180#405—172no—
181#405—172no—
182C4×M16—172no—
183C4×M16#59511yescharacteristic
184C2×C4×C8—171no—
185C2×C4×C8—172no—
186C4⋊C16—171no—
187C4⋊C16—171no—
188C4⋊C16—172no—
189C4⋊C16—172no—
190C4⋊C16—172no—
191C4⋊C16—174no—
192C4⋊C16—174no—
193C2×C17—22no—
194C2×C17C22×C4⋊C811yescharacteristic
195C2×C17C2×#50411yescharacteristic
196C2×C17#340711yescharacteristic
197D17—24no—
198C2×#41—171no—
199#131—174no—
200#131—174no—
201#131—344no—
202C2×M16—172no—
203C2×M16—172no—
204C2×M16C2×#59511yescharacteristic
205C2×M16C2×#59511yescharacteristic
206C22×C8—171no—
207C22×C8—172no—
208C22×C8—171no—
209C22×C8—172no—
210C4×Q8—171no—
211C4×Q8—171no—
212C4×D4—172no—
213C4×D4—172no—
214C4×D4—171no—
215C4×D4—171no—
216#117—171no—
217#117—172no—
218C2×C42—172no—
219C2×C42C2×#59511yescharacteristic
220M32—348no—
221M32—344no—
222C2×C16—172no—
223C2×C16—172no—
224C2×C16—174no—
225C2×C16—348no—
226C2×C16—344no—
227#97—172no—
228#97—172no—
229#97C2×#59512yes—
230C4×C8—172no—
231C4×C8—172no—
232C4×C8—171no—
233C4×C8—171no—
234C4×C8—171no—
235C4×C8—171no—
236C4×C8C2×#59512yes—
237C17#3339811yescharacteristic, complemented, Sylow
238#41—174no—
239#41—174no—
240C2×Q8—171no—
241C2×D4—172no—
242C2×D4—171no—
243C22×C4—172no—
244C22×C4—172no—
245C22×C4—172no—
246C22×C4#6324211yescharacteristic
247C22×C4#6324211yescharacteristic
248C22×C4C22×#59511yescharacteristic
249M16—22no—
250M16—178no—
251M16—344no—
252M16C4⋊#59514yes—
253C2×C8—172no—
254C2×C8—172no—
255C2×C8—171no—
256C2×C8—171no—
257C2×C8—174no—
258C2×C8—174no—
259C2×C8—171no—
260C2×C8—171no—
261C2×C8—344no—
262C2×C8—342no—
263C2×C8C22×#59512yes—
264C2×C8C22×#59512yes—
265C4⋊C4—172no—
266C4⋊C4—172no—
267C4⋊C4—171no—
268C4⋊C4—171no—
269C22⋊C4—172no—
270C22⋊C4—172no—
271C22⋊C4—172no—
272C42—172no—
273C42—172no—
274C42—172no—
275C42#6324211yescharacteristic
276C42#6324211yescharacteristic
277C42C22×#59511yescharacteristic
278C42C22×#59511yescharacteristic
279C16—342no—
280C16—344no—
281C16—342no—
282C23—172no—
283C23C2×#6324211yescharacteristic
284Q8—174no—
285D4—178no—
286D4—174no—
287C2×C4—22no—
288C2×C4—41no—
289C2×C4—172no—
290C2×C4—174no—
291C2×C4—174no—
292C2×C4—172no—
293C2×C4—172no—
294C2×C4—344no—
295C2×C4—682no—
296C2×C4C2×#6324211yescharacteristic
297C2×C4C2×#6324211yescharacteristic
298C2×C4C2×#6324211yescharacteristic
299C2×C4C2×#6324211yescharacteristic
300C2×C4C2×#6324211yescharacteristic
301C2×C4D17⋊M1612yes—
302C2×C4C2×C4⋊#59512yes—
303C2×C4C23×#59511yescharacteristic
304C8—22no—
305C8—171no—
306C8—171no—
307C8—172no—
308C8—341no—
309C8—341no—
310C8C2×C4⋊#59514yes—
311C22—21no—
312C22—21no—
313C22—174no—
314C22—342no—
315C22—342no—
316C22#1176381012yes—
317C22C22×#6324211yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—174no—
322C4—342no—
323C4—342no—
324C4#1176381012yes—
325C4#1176386812yescomplemented
326C4C22×C4⋊#59511yescharacteristic
327C4C2×D17⋊M1611yescharacteristic
328C2—22no—
329C2—344no—
330C2C2×#1176381011yescharacteristic
331C2C2×#1176386811yescharacteristic
332C2#1454575311yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×#595
commutator, socleC22×C17C23×C8
FittingC4×M16×C17C4
prime radicalC2×#63252C2×C4
cosocleC2×#3622C24
hypercenterC4×M16#595
hypocenterC17#33398
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C4×M16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #14545121C2semidirect
3. #14545120C2semidirect
4. #14545117C2semidirect
5. #14545116C2semidirect
6. #14545114C2semidirect
7. C2×C4⋊#63216C2semidirect
164. C4×C17#504semidirect
237. C17#33398semidirect
325. C4#11763868semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#117638682
#145451212
#145451202
#145451172
#145451162
#145451142
C2×C4⋊#632162
C4×#105875932
#3339817